Now let me shift the frequency a little bit, just to see what it looks like. So to
shift that center frequency. In other words, off zero because we haven't done
anything off zero yet. We'll just go over here, and say, ten. Alright, now you may
go like, what happened? Alright? Well I'll tell you what happened. Moment here, this
goes up. So first of all, I just shifted the center frequency over by ten, still
fixed in frequency. Now the top picture is interesting. Now I am looking at this, my
center frequency is zero and zero frequency, what does that mean? Cosine
zero X, n oscillations. Now my center frequency is cosine ten X, well cosine ten
X. I mean I have this, I have this e to the I ten X multiplying everything. So you
see this here? Looks like a cosine 10x sitting on that thing. So I have a bump,
this hyperbolic secant times the cosine ten X. So it's a wave packet, oscillation.
So if you want to get rid of it, you'd say, well how about if I just look at the
absolute value of that thing? And if you do that, it takes away the phase, and it
gives you back that. Okay, all that oscillation was, was having a group of
carrier frequencies sitting around in there. It's a frequency component and it
has a phase. You just get rid of them. Absolute value, stuff like that. It's all
I care about. You may care about the phase for more important reasons later, but for
right now, it's all we care about. Notice a couple things, if I had a lot of noise
on here. I'm going to bury all of this so in fact let's, let's, let's bury all of it
and just put it in a sea of noise and your going to see that. Basically, if I take
all of this out and this is all noisy then what do I do. Okay, first thing you gotta
do, if you have this all buried in noise, figure out where is this, where's that
center frequency. Second thing you do is, once you know where the center frequency
is, you can build the filter there, and build the filter so you can go back to
here, okay. Alright, so let's bury the noise. And to bury it in noise is, is
amazingly eas y to do, right? So you have your function, and there it is, and all
you got to do bury it in noise is, for every frequency component you got. Let me
see where this ends. Here, so I'm FFT'ing the signal right? And all I gotta do is
add, plus noise, times rand N um., this is one by N, plus I times rand N, one by N.
That's it, so like we did before except I haven't told you what noise is. So let's
just do noise is equal to like, twenty. I'll just put a bunch of noise on it and
let's look at this thing. Whoa, tell me if you see anything in there okay now that
really bug becomes a real shit alright whatever or shit test right. Roshard
whatever guy you see in there? Keon do you see anything in there no nothing dog cat
your house you grew up in no nothing carpet. Okay, anybody else, want to
volunteer anything else, better than carpet, no, okay. All right, by the way,
this is what you got, to work with, right? And here's what a my claim is, if I keep
taking samples of this thing, in fact, I have this thing, I have 21 samples, right,
I haven't made any use of that fact. Right, I've 21 samples, these are all
white noise, and in there is just this clear line, that you cannot see. There is
a signal sitting there, waiting for you to go get it. Okay, now lets talk about, if
you have this, what does this do to here? So that's what it looks like here what
does it look like in the time domain. And not so hard to do you gotta inverse before
you transform everything else you call that UN. all I gotta do I've gotta
interverse 4A transform it. So I have to go, basically, now if you shoot. I FFT
shifted it all ahead of time right into the absolute value. So, let's go and do
some of that. Let's take out the absolute value. Let's take off that FFT shift.
Okay, all that hard work thrown away, fine we'll, we'll, we'll FFT shift it later.
All we have to do is take the absolute value here and FFT shift it there instead.
Okay, but U of N now is just the IFFT of U, J. Okay, O UT sorry yeah. So I take row
by row this noisy signal. I inversef ully transform it, and now I can say supply 211
and we can just plot water fall TS absolute value of U and, Oh I need my view
back. Okay, so here this is your problem. Tell me if there's anything there. So
remember data analysis is about this. So what, what I like about these examples is
in your dog problem or in this here. You clearly saw, we already know the answer
ahead of time. Right? We know there's a signal in there. We know there's this
thing going like this through here. You can't see it cause I hit it, right? And
the objective, if you are the airplane is to hide yourself. The objective if you are
there in that radar is to, is, is hide and go seek, right, and but it's, it's whoever
has got the better, better data analysis processing skills, right? There are things
here you have to take advantage of. If you don't take advantage of them, you get no
information, okay? And what's the information you could take advantage of?
So first of all in this problem. You may not even know where a filter exist put.
What happens if this airplane bounces off your omeganaut? Suppose, okay? Let, let,
let's play a what if game. Suppose you develop really great stealth technology.
This plane here you know how to put a, a nice coating around your, your airplane.
Design it so that you send omeganaut out. Nothing comes back. Now, if you're these
guys, you're like, wow, that's plane visible to me. However what if you're
emitting some other frequency? There's something else coming off your plane off
the engine it wasn't something I shot out, shot out there, but I have this big
spectrum of stuff that I'm reading in. Maybe you're emitting at some other
frequency some information that I could get. I don't even know what frequency I
should be lookin for. But if I keep taking measurements what this is going to allow
me to do is come back to this idea here, is to say, here's what I'm picking up
which just looks like nothing. And what I want to start doin, slice by slice, is I
just keep taking information in, start averaging it. Because anything that's real
signal is not going to cancel out. Noise should cancel out, right? So then if I get
all the noise cancellation happening. Okay, sorry for the pause. Alright, ready?
So if I get all the noise cancelling out, I will start to see emerging out of this
frequency component a signature there, somewhere. And maybe I don't know where it
is ahead of time, and if I see it. Then you can say well, you didn't put a filter
around here, get rid of everything else put the filter there, see what it says
over here. See if there's a signal there, right. So this is the kind of stuff you
would want to do. It's all about who's better at signal processing actually, at
the end of the day. Alright, so I've given you this example here which is, sort of
just using very basic time frequency. Which is I can represent the whole signal
in time or I can represent it in frequency. Now what we're going to move
onto is much more sophisticated thinking. Which is to say, wait a minute, somehow.
When I go from time to frequency, if I'm in the frequency domain, it tells me
nothing about the time domain. If I'm in the time domain, it tells me nothing about
frequency. I want both pieces of information at the same time. And this is
where time/frequency analysis comes in which is to say, I would trade a little
bit of what I know in the time domain for a little information on the frequency
domain. I'm not going to give up everything I know in the time domain.
They're not separate objects. They, I'm going to use both at the same time, okay?
Remember you can not know everything about time and frequency at the same time.
Impossible, through the idea of the Fourier transform, the idea of Heisenberg
uncertainty. That's not a possibility for you. However, you can sacrifice one for
the other to try to get yourself in a situation where you can know a little bit
about both and maybe make some information there. Okay? Alright, so Wednesday, we're
going to talk about window Fourier transforms, which is this idea of how do I
represent both. In the meantime, you have your dog problem, and we'll just keep
moving forward from there.